Is observational congruence on μ-expressions axiomatisable in equational Horn logic?

It is well known that bisimulation on μ-expressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom/Ésik necessarily involve implicational rules. However, both systems rely on features beyond pure equational Horn logic: either rules that are impure by involving non-equational side-conditions, or rules that are schematically infinitary like the congruence rule which is not Horn. It is an open question whether these complications cannot be avoided in the proof-theoretically and computationally clean and powerful setting of second-order equational Horn logic.This paper presents a positive and a negative result regarding the axiomatisability of observational congruence in equational Horn logic. Firstly, we show how Milner’s impure rule system can be reworked into a pure Horn axiomatisation that is complete for guarded processes. Secondly, we prove that for unguarded processes, both Milner’s unique fixed point rule and Bloom/Ésik’s GA rule are incomplete without the congruence rule, and neither system has a complete extension in rank-1 equational axioms. It remains open whether there are higher-rank equational axioms or other Horn rules which would render Milner’s or Bloom/Ésik’s axiomatisations complete.